Show that the middle term in the expansion of(1+x)raise to power 2n is = 1.3.5.......(2n-1) . 2n.xraise to power n upon n! , where nis a +ve integer.

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Total no of terms are (2n+1)  which are odd (right?), so middle term will be (2n+1 +1) /2  th term i.e. (n+1)th term

Now middle term i.e. (n+1) th term of (1+x)2n = 2nCn (x)n,  = !2n /(!n *!n) = (2n)xn/ (!n * !n) 

(remember 1.2.3.....(2n) are total '2n' terms out of which 'n' will be even (i.e. 2.4.6.....upto (2n)  ) and 'n' are odd (i.e. (2n-1)  ). (2n)/ (!n * !n) = * (2.4.6.....2n)xn /!n *!n

Taking out '2' each from these 'n'  'even' termswe get  1.3.5...(2n-1) (2n)( xn/!n *!n,

or ,1.3.5...(2n-1) (2n) !n/(!n*!n) which gives 1.3.5...(2n-1) (2n)xn/ !n

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